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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brtxpsd | Structured version Visualization version GIF version | ||
| Description: Expansion of a common form used in quantifier-free definitions. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
| Ref | Expression |
|---|---|
| brtxpsd.1 | ⊢ 𝐴 ∈ V |
| brtxpsd.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| brtxpsd | ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-br 5144 | . . 3 ⊢ (𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ 〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V))) | |
| 2 | opex 5469 | . . . . 5 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 3 | 2 | elrn 5904 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉) |
| 4 | brsymdif 5202 | . . . . . 6 ⊢ (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ¬ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉)) | |
| 5 | brv 5477 | . . . . . . . . 9 ⊢ 𝑥V𝐴 | |
| 6 | vex 3484 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
| 7 | brtxpsd.1 | . . . . . . . . . 10 ⊢ 𝐴 ∈ V | |
| 8 | brtxpsd.2 | . . . . . . . . . 10 ⊢ 𝐵 ∈ V | |
| 9 | 6, 7, 8 | brtxp 35881 | . . . . . . . . 9 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ (𝑥V𝐴 ∧ 𝑥 E 𝐵)) |
| 10 | 5, 9 | mpbiran 709 | . . . . . . . 8 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥 E 𝐵) |
| 11 | 8 | epeli 5586 | . . . . . . . 8 ⊢ (𝑥 E 𝐵 ↔ 𝑥 ∈ 𝐵) |
| 12 | 10, 11 | bitri 275 | . . . . . . 7 ⊢ (𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥 ∈ 𝐵) |
| 13 | brv 5477 | . . . . . . . 8 ⊢ 𝑥V𝐵 | |
| 14 | 6, 7, 8 | brtxp 35881 | . . . . . . . 8 ⊢ (𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉 ↔ (𝑥𝑅𝐴 ∧ 𝑥V𝐵)) |
| 15 | 13, 14 | mpbiran2 710 | . . . . . . 7 ⊢ (𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉 ↔ 𝑥𝑅𝐴) |
| 16 | 12, 15 | bibi12i 339 | . . . . . 6 ⊢ ((𝑥(V ⊗ E )〈𝐴, 𝐵〉 ↔ 𝑥(𝑅 ⊗ V)〈𝐴, 𝐵〉) ↔ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
| 17 | 4, 16 | xchbinx 334 | . . . . 5 ⊢ (𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
| 18 | 17 | exbii 1848 | . . . 4 ⊢ (∃𝑥 𝑥((V ⊗ E ) △ (𝑅 ⊗ V))〈𝐴, 𝐵〉 ↔ ∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
| 19 | 3, 18 | bitri 275 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ ran ((V ⊗ E ) △ (𝑅 ⊗ V)) ↔ ∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
| 20 | exnal 1827 | . . 3 ⊢ (∃𝑥 ¬ (𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴) ↔ ¬ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) | |
| 21 | 1, 19, 20 | 3bitrri 298 | . 2 ⊢ (¬ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴) ↔ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵) |
| 22 | 21 | con1bii 356 | 1 ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑅 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑅𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∀wal 1538 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 △ csymdif 4252 〈cop 4632 class class class wbr 5143 E cep 5583 ran crn 5686 ⊗ ctxp 35831 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-symdif 4253 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 |
| This theorem is referenced by: brtxpsd2 35896 |
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